Assumptions on one-way functions

Question

What are the assumptions we make to propose a one-way function? I only know of some number theoretic assumptions, but what are the other assumptions you can possibly make, which if true, would imply that $f$ is a one-way function?

(Considering pre-image resistant hash functions, what is the assumption there? Hashes as far as I know do not rely on any assumption other than the fact that it seems to be hard)

Edit:

Example: Say we assume factoring a semiprime $N = pq$ is hard. Then we can say that the function $f(p,q) = p*q$ is one-way. What other assumptions exist?

Answer 1

I'd recommend reading Barak's recent survey titled "The Complexity of Public-Key Cryptography". The survey explores the assumptions required for one-way functions and public-key cryptography (loosely speaking, the latter requires more structured assumptions compared to the former). There (in §2) you can find several candidate constructions of one-way functions from diverse problems arising from cryptography (e.g., stream ciphers, hash functions), complexity theory (e.g., the planted SAT and planted Clique problem), combinatorics (e.g., Goldreich's one-way function) and learning theory, and he argues why these are plausibly one-way.

Answer 2

General

The existence of a "perfect" one-way function is not proven. Such a function would prove that P $\ne$ NP, which is still an unsolved problem in mathematics / computer science.

One-way functions are per definition hard to invert, but easy to check. One example, that are assumed that they are one-way, are hash-functions:

You can easily create a hash of a given input, but you can't get the input from a given hash in a reasonable amount of time, assuming the input is long enough.

Some properties of a one-way function:

• Easy to compute, but infeasible to compute the inverse of the function in the average case
• Preimage resistant

Hashes

We assume that there exists a perfect one-way hash function: Hash functions are a special kind of one-way functions, since they reduce a possible large input to a fixed length smaller hash value, the term is a one-way compression function. This means that there is a data-loss and also means that mulitple different inputs can have the same hash value (pigeonhole principle). Data-loss here means, that it's impossible to retrieve the unknown original input from a given hash-value.